Designing a Resilient Allee-Ornstein-Uhlenbeck model

TL;DR

This paper develops a stochastic control framework for an Allee-OU model, stabilizing population dynamics under environmental and demographic noise to prevent extinction.

Contribution

It introduces a novel control approach for an Allee-OU model, enabling stabilization of population fluctuations through stationary distribution approximation.

Findings

Control stabilizes population around equilibrium

Threshold-based interventions reduce intervention frequency

Simulations confirm effective fluctuation mitigation

Abstract

In stochastic population dynamics, stochastic wandering can produce transition to an absorbing state. In particular, under Allee effects, low densities amplify the possibility of population collapse. We investigate this in an Allee-Ornstein-Uhlenbeck (Allee-OU) model, that couples a bistable Allee growth equation, with demographic noise, and environmental fluctuations modeled as an Ornstein-Uhlenbeck process. This process replaces the bifurcation parameter of the deterministic Allee effect equation. In the model, small noise may induce escape from the safe basin around the positive equilibrium toward extinction. We construct a stochastic control, altering the process to have a stationary distribution. We enable tractable control design, approximating the process by one with a stationary distribution. Two controlled models are developed, one acting directly on population size and another also modulating the environment. A threshold-based implementation minimizes the frequency of interventions while maximizing safe time. Simulations demonstrate that the control stabilizes fluctuations around the equilibrium.